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Generalized bitangent Caratheodory–Nevanlinna–Pick problem, and $(j,J_0)$-inner
matrix-valued functions
D. Z. Arov
Abstract:
This paper is a study of the problem of describing holomorphic
$n\times n$ matrix-valued functions
$c(z)$ on the unit disk
$K$ with
$\operatorname{Rec}(z)\geqslant 0$ (the Caratheodory class
$\mathbf C_n$) such that
$b_1^{-1}(c-c_0)b_2^{-1}\in\mathscr D_n$, where
$b_1$,
$b_2$, and
$c_0$ are particular matrix-valued functions with
$b_1$ and
$b_2$ inner and
$c_0$ in
$\mathbf C_n$, and
$\mathscr D_n$ is the Smirnov class of matrix-valued functions of bounded type on
$K$. The matrix extrapolation problems of Caratheodory, Nevanlinna–Pick, and M. G. Krein reduce to this problem for special
$b_1$ and
$b_2$, as do even the tangent and
$*$-tangent problems when there is extrapolation data for
$c(z)$ and
$c^*(z)$ not on the whole Euclidean space
$C^n$ but only on chains of its subspaces. In the completely indeterminate case the solution set of the problem is obtained as the image of the class
$B_n$ of holomorphic contractive
$n\times n$ matrix-valued functions on
$K$ under a linear fractional transformation with
$(j,J_0)$-inner matrix-valued function
$A(z)=[a_{ik}(z)]_1^2$ of coefficients on
$K$. The
$A(z)$ arising in this way form a class of regular
$(j,J_0)$ -inner matrix-valued functions whose singularities appear to be determined by the singularities of
$b_1$ and
$b_2$. The general results are applied to Krein's problems of extension of helical and positive-definite matrix-valued functions from a closed interval.
UDC:
517.5
MSC: Primary
30E05,
30D05,
30D50; Secondary
47A56,
47A57,
15A22 Received: 28.11.1991